= Solution
Put $\xi=x-vt$ and $U(\xi)=A\operatorname{sech}^2\xi$. For $A\ne0$, differentiation gives $U''=4U-6U^2/A$ and therefore $U^{(3)}=4U'-(12/A)UU'$. The <Korteweg-De Vries equation> reduces to
$$
(4-v)U'-\left(6+\frac{12}A\right)UU'=0.
$$
Since $U$ is nonconstant, both coefficients must vanish. Thus the nonzero <solitary wave> is
$$
\boxed{A=-2,\qquad v=4,\qquad u=-2\operatorname{sech}^2(x-4t).}
$$
There is also the trivial solution $A=0$, for which $v$ has no effect. The negative amplitude follows from the minus sign of the nonlinear term; it agrees with the <sech-squared solitary wave of the KdV equation>.
\Image[/past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2008/paper-2-negative-kdv-soliton.png]
{title=A negative solitary wave translates to the right without changing its profile}
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